Quantum conditional operator and a criterion for separability

TL;DR

This paper introduces a new criterion for quantum state separability based on the properties of the conditional amplitude operator and a positive map, providing both necessary and sufficient conditions in certain low-dimensional systems.

Contribution

It presents a novel separability criterion using the conditional amplitude operator and a positive map, linking it to partial transposition and time-reversal for specific quantum systems.

Findings

Conditional amplitude operator eigenvalues are invariant under local unitaries.

Separable states cannot have eigenvalues of the conditional operator exceeding 1.

The proposed criterion is equivalent to partial transposition for 2x2 and 2x3 systems.

Abstract

We analyze the properties of the conditional amplitude operator, the quantum analog of the conditional probability which has been introduced in [quant-ph/9512022]. The spectrum of the conditional operator characterizing a quantum bipartite system is invariant under local unitary transformations and reflects its inseparability. More specifically, it is shown that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. This leads us to consider a related separability criterion based on the positive map $\Gamma:\rho \to (Tr \rho) - \rho$, where $\rho$ is an Hermitian operator. Any separable state is mapped by the tensor product of this map and the identity into a non-negative operator, which provides a simple necessary condition for separability. In the special case where one subsystem is a quantum bit, $\Gamma$ reduces to time-reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\times 2$ and $2\times 3$ systems. Finally, a simple connection between this map and complex conjugation in the "magic" basis is displayed.